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2 years ago

Help me to answer this pls

Mathematics

1 answer:

zepelin [54]2 years ago

8 0

The 6 angles with vertex P are:

Angle APB
Angle APC
Angle APD
Angle BPC
Angle BPD
Angle CPD

The letter P must be in the middle since this location is the vertex. Something like angle BPA is the same as angle APB. We can swap the order of the outer letters without any change to the angle itself. Effectively, this means each angle has two possible names.

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I got ALL of the answers but I can't explain it at all... please explain guys I have NO idea!

nekit [7.7K]

Explanation:

The altitude CH divides triangle ABC into similar triangles:

ΔABC ~ ΔACH ~ ΔCBH

Angle bisector AL divides the triangle(s) into proportional parts:

BL/BA = CL/CA

HD/HA = CD/CA

Of course, the Pythagorean theorem applies to the sides of each right triangle:

AH^2 +CH^2 = AC^2

DH^2 +AH^2 = AD^2

LC^2 + AC^2 = LA^2

AC^2 +BC^2 = AB^2

And segment lengths sum:

HD +DC = HC

AD +DL = AL

AH +HB = AB

CL +LB = CB

Solving the problem involves picking the relations that let you find something you don't know from the things you do know. You keep going this way until the whole geometry is solved (or, at least, the parts you care about).

___

We can use the Pythagorean theorem to find AH right away, since we already know AD and DH.

DH^2 +AH^2 = AD^2

4^2 + AH^2 = 8^2 . . . . . . . substitute known values

AH^2 = 64 -16 = 48 . . . . . . subtract 16

AH = 4√3 . . . . . . . . . . . . . . take the square root

Now, we can use this with the angle bisector relation to tell us how CD and CA are related.

HD/HA = CD/CA

4/(4√3) = CD/CA . . . . . substitute known values

CA = CD·√3 . . . . . . . . . cross multiply and simplify

Using the sum of lengths equation, we have ...

CH = HD +CD

CH = 4 + CD

From the Pythagorean theorem ...

AH^2 +CH^2 = AC^2

(4√3)^2 + (4 +CD)^2 = (CD√3)^2 . . . . . substitute known values

48 + (16 +8·CD +CD^2) = 3·CD^2 . . . . . simplify a bit

2·CD^2 -8·CD -64 = 0 . . . . . . . . . . . . . . . put the quadratic into standard form

2(CD -8)(CD +4) = 0 . . . . . . . . . . . . . . . . factor

CD = 8 . . . . . only the positive solution is useful here

Now, we know ...

∆ADC is isosceles, so ∠ACH = ∠CAD = ∠DAH = ∠CBA

CH = 8+4 = 12

AC = 8√3 . . . . . = 2·AH

Then by similar triangles, ...

AB = 2·AC = 16√3

BC = AC·√3 = 24

7 0

3 years ago

A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 25 fe

Varvara68 [4.7K]

Answer:

$Length =\frac{25}{4 + \pi}$ and $Width = \frac{50}{4+\pi}$

Step-by-step explanation:

This question is better understood with an attachment.

See attachment for illustration.

Given

Represent Perimeter with P

$P = 25ft$

Required

Determine the dimension of the rectangle that maximizes the area

First, we calculate the perimeter of the rectangular part of the window.

From the attachment, the rectangle is not closed at the top.

So, The perimeter would be the sum of the three closed sides

Where

$Width = 2x$

$Length = y$

So:

$P_{Rectangle} = y + y + 2x$

$P_{Rectangle} = 2y + 2x$

Next, we determine the circumference of the semi circle.

Circumference of a semicircle is calculated as:

$C = \frac{1}{2}\pi r$

From the attachment,

$Radius (r) = x$

So, we have:

$C = \frac{1}{2}2\pi * x$

$C = \pi x$

So, the perimeter of the window is:

$P = P_{Rectangle} + C$

$P =2y + 2x + \pi x$

Recall that: $P = 25$

So, we have:

$25 =2y + 2x +\pi x$

Make 2y the subject

$2y = 25 - 2x - \pi x$

Make y the subject:

$y = \frac{25}{2} - \frac{2x}{2} - \frac{\pi x}{2}$

$y = \frac{25}{2} - x - \frac{\pi x}{2}$

Next, we determine the area (A) of the window

A = Area of Rectangle + Area of Semicircle

$A = 2x * y + \frac{1}{2}\pi r^2$

$A = 2xy + \frac{1}{2}\pi r^2$

Recall that

$Radius (r) = x$

$A = 2xy + \frac{1}{2}\pi x^2$

Substitute $\frac{25}{2} - x - \frac{\pi x}{2}$ for y in $A = 2xy + \frac{1}{2}\pi x^2$

$A = 2x(\frac{25}{2} - x - \frac{\pi x}{2}) + \frac{1}{2}\pi x^2$

Open Bracket

$A = 2x * \frac{25}{2} - 2x * x - 2x * \frac{\pi x}{2} + \frac{1}{2}\pi x^2$

$A = 25x - 2x^2 - \pi x^2 + \frac{1}{2}\pi x^2$

$A = 25x - 2x^2 - \frac{1}{2}\pi x^2$

To maximize area, we have to determine differentiate both sides and set A' = 0

Differentiate

$A' = 25 - 4x - \pi x$

$A' = 0$

So, we have:

$0 = 25 - 4x - \pi x$

Factorize:

$0 = 25 -x(4 + \pi)$

$-25 =-x(4 + \pi)$

Solve for x

$x = \frac{-25}{-(4+\pi)}$

$x = \frac{25}{4+\pi}$

Recall that

$Width = 2x$

$Width = 2(\frac{25}{4+\pi})$

$Width = \frac{50}{4+\pi}$

Recall that:

$y = \frac{25}{2} - x - \frac{\pi x}{2}$

Substitute $\frac{25}{4+\pi}$ for x

$y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{\pi (\frac{25}{4+\pi})}{2}$

$y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{\frac{25\pi}{4+\pi}}{2}$

$y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{25\pi}{4+\pi} * \frac{1}{2}$

$y = \frac{25}{2} - \frac{25}{4+\pi} - \frac{25\pi}{2(4+\pi)}$

$y = \frac{25(4+\pi) - 25 * 2 - 25\pi}{2(4 + \pi)}$

$y = \frac{100+25\pi - 50 - 25\pi}{2(4 + \pi)}$

$y = \frac{100- 50+25\pi - 25\pi}{2(4 + \pi)}$

$y = \frac{50}{2(4 + \pi)}$

$y = \frac{25}{4 + \pi}$

Recall that:

$Length = y$

So:

$Length =\frac{25}{4 + \pi}$

Hence, the dimension of the rectangle is:

$Length =\frac{25}{4 + \pi}$ and $Width = \frac{50}{4+\pi}$

3 0

3 years ago

Write a system of linear equations that has the ordered pair as its solution (1,4)

Masteriza [31]

Since the only information given is that the order pair (1,4) is a solution of the system of linear, we are therefore having an infinite number of options.
The trivial equations would be:
x = 1
y = -4

We can write equations combining both variables as:
y = 4x
y = 3x + 1
y = 2x + 2
y = x + 3
y = -4x + 8
y = -3x + 7
y = -2x + 6
y = -x + 5

All the above can be used as linear equations of the line and would have (1,4) as a solution.
In order to define the exact line, we need at least two points, which are not given in this question.

8 0

4 years ago

The diameter of a circle is 6 in. What is the circle's circumference?

likoan [24]

Answer: 18.84

Step-by-step explanation:

5 0

3 years ago

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Christine has nine shining rings. If six of them are gold, how many are not gold?

Anastasy [175]

3 of them would not be good